## On linear Volterra equations in Banach spaces.(English)Zbl 0569.45020

The authors study the linear equation $$u'(t)=Au(t)+\int^{t}_{0}B(t- s)u(s)ds+f(t),\quad u(0)=u_ 0,$$ in a Banach space X. They prove that there exists a reasonable resolvent operator if and only if the autonomous equation (where $$f=0)$$ is well-posed (i.e., it has a unique solution that depends continuously on $$u_ 0)$$. Furthermore, under some additional weak restrictions they show that a necessary and sufficient condition for this to happen is that $$| (1/n!)H^{(n)}(\lambda)| \leq M(Re \lambda -\omega)^{-n-1}$$ for all Re $$\lambda$$ $$>\omega$$, $$n\geq 0$$, where $$H(\lambda)=(\lambda -A-\hat B(\lambda))^{-1},$$ that is a result of Hille-Yosida type. The authors also give an example showing that this condition can be satisfied although A does not generate a semigroup.
Reviewer: G.Gripenberg

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45D05 Volterra integral equations

### Keywords:

Volterra equation; Banach space; resolvent; well-posed; semigroup
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### References:

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